# What is the Cartesian form of (36,pi)?

##### 1 Answer
Jun 28, 2016

$\left(- 36 , 0\right)$

#### Explanation:

The formulas used to convert between polar and Cartesian co-ordinates are:
$x = r \cos \theta$
$y = r \sin \theta$

We are given the polar co-ordinate point $\left(r , \theta\right) \to \left(36 , \pi\right)$, so $r = 36$ and $\theta = \pi$. Making substitutions in the above formulas:
$x = 36 \cos \left(\pi\right) = 36 \cdot \left(- 1\right) = - 36$
$y = 36 \sin \left(\pi\right) = 36 \left(0\right) = 0$

Therefore, the point in the Cartesian plane is $\left(- 36 , 0\right)$.

Intuitively, this result makes sense because $\pi$ is a half-circle rotation, so in the Cartesian plane an angle of $\pi$ corresponds to a point on the $x$-axis (and therefore $y = 0$). A radius of $36$ means the point is $36$ units to the left or right of the origin, and since an angle of $+ \pi$ is a counterclockwise rotation, it would mean the point is $36$ units to the left (and therefore $- 36$).